prove $|X-Y|\le|X|+|Y|$

Question

MY question is how to prove $|X-Y|\le|X|+|Y|$.

My textbook proves another version of triangular inequality, which is

$|A|-|B|\le|A-B|$ by substituting X=A-B and Y=B into $|X+Y|\le|X|+|Y|$,

but it leaves the above version as an exercise, and I can't solve it.

Thank you in advance. :)

Answer 1

By the triangle inequality $$|X|+|Y|=|X|+|-Y|\geq|X+(-Y)|=|X-Y|.$$